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Kato's Conjecture
Kato's conjecture is a mathematical problem named after mathematician Tosio Kato, of the University of California, Berkeley. Kato initially posed the problem in 1953. Kato asked whether the square roots of certain elliptic operators, defined via functional calculus, are analytic. The full statement of the conjecture as given by Auscher ''et al.'' is: "the domain of the square root of a uniformly complex elliptic operator L =-\mathrm (A\nabla) with bounded measurable coefficients in Rn is the Sobolev space ''H''1(Rn) in any dimension with the estimate , , \sqrtf, , _ \sim , , \nabla f, , _". The problem remained unresolved for nearly a half-century, until in 2001 it was jointly solved in the affirmative by Pascal Auscher, Steve Hofmann Steve Hofmann is a mathematician who helped solve the famous Kato's conjecture. Said Hofmann, “It's a problem that has interested me since I was a graduate student... It was one of the biggest open problems in my field and everybody thought it wa . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Problem
A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. Real-world problems Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems, they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics. In general, to use mathematics for solving a real-world problem, the first ste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tosio Kato
was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis. Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption of the Second World War, he received his doctorate in 1951 from the University of Tokyo, where he became a professor in 1958. From 1962, he worked as a professor at the University of California at Berkeley in the United States. Many works of Kato are related to mathematical physics. In 1951, he showed the self-adjointness of Hamiltonians for realistic (singular) potentials. He dealt with nonlinear evolution equations, the Korteweg–de Vries equation (Kato smoothing effect in 1983) and with solutions of the Navier-Stokes equation. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant university and the founding campus of the University of California system. Its fourteen colleges and schools offer over 350 degree programs and enroll some 31,800 undergraduate and 13,200 graduate students. Berkeley ranks among the world's top universities. A founding member of the Association of American Universities, Berkeley hosts many leading research institutes dedicated to science, engineering, and mathematics. The university founded and maintains close relationships with three national laboratories at Berkeley, Livermore and Los Alamos, and has played a prominent role in many scientific advances, from the Manhattan Project and the discovery of 16 chemical elements to breakthroughs in computer science and genomics. Berkeley is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_n^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x^2 and M an n\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Auscher
Pascal Auscher is a French mathematician working at University of Paris-Sud. Specializing in harmonic analysis and operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ..., he is mostly known for, together with Steve Hofmann, Michael Lacey, Alan McIntosh and Philippe Tchamitchian, solving the famous Kato's conjecture. References External links * Living people Year of birth missing (living people) 21st-century French mathematicians {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steve Hofmann
Steve Hofmann is a mathematician who helped solve the famous Kato's conjecture. Said Hofmann, “It's a problem that has interested me since I was a graduate student... It was one of the biggest open problems in my field and everybody thought it was too hard and wouldn't be solved. I had toyed with it for years and then put in three years of very serious work before hitting the key breakthrough.â Hofmann, Curators' professor at the University of Missouri, worked alongside other prominenent mathematicians (Pascal Auscher, Michael Lacey, John Lewis, Alan McIntosh and Philippe Tchamitchian) to solve this problem, one that was put into place in the early 1950s by Tosio Kato, a Mathematician at The University of California at Berkeley. Hofmann received his PhD from the University of Minnesota, Twin Cities. He delivered an invited address at the 2006 International Congress of Mathematicians in Madrid. In 2012 he became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Lacey (mathematician)
Michael Thoreau Lacey (born September 26, 1959) is an American mathematician. Lacey received his Ph.D. from the University of Illinois at Urbana-Champaign in 1987, under the direction of Walter Philipp.. His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated logarithm for empirical characteristic functions. In the intervening years, his work has touched on the areas of probability, ergodic theory, and harmonic analysis. His first postdoctoral positions were at the Louisiana State University, and the University of North Carolina at Chapel Hill. While at UNC, Lacey and Walter Philipp gave their proof of the almost sure central limit theorem. He held a position at Indiana University from 1989 to 1996. While there, he received a National Science Foundation Postdoctoral Fellowship, and during the tenure of this fellowship he began a study of the bilinear Hilbert transform. This transform was at the time the subject of a conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Gaius Ramsay McIntosh
Alan Gaius Ramsay McIntosh (* 1942 in Sydney, †August 8, 2016 ) was an Australian mathematician who dealt with analysis (harmonic analysis, partial differential equations). He was a professor at the Australian National University in Canberra. McIntosh studied at the University of New England with a bachelor's degree in 1962 (as a student he also received the University Medal ) and PhD in 1966 with Frantisek Wolf at the University of California, Berkeley, ( Representation of Accretive Bilinear Forms in Hilbert Space by Maximal Accretive Operator ). In Berkeley, he was also a student of Tosio Kato. As a post-doctoral student, he was at the Institute for Advanced Study and from 1967 he taught at Macquarie University and from 1999 at the Australian National University. In 2014 he became emeritus. McIntosh was involved in solving the Calderon conjecture in the theory of singular integral operators. In 2002, he solved with Pascal Auscher, Michael T. Lacey, Philipp Tchamitchia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philippe Tchamitchian
Philippe is a masculine sometimes feminin given name, cognate to Philip. It may refer to: * Philippe of Belgium (born 1960), King of the Belgians (2013–present) * Philippe (footballer) Philippe Almeida Costa (born 1 March 2000) is a Brazilian professional footballer who plays as a forward. Professional career Batata began playing senior football with Vila Nova Futebol Clube in the Campeonato Goiano in 2018. On 16 May 2018, Phi ... (born 2000), Brazilian footballer * Prince Philippe, Count of Flanders, father to Albert I of Belgium * Philippe d'Orléans (other), multiple people * Philippe A. Autexier (1954–1998), French music historian * Philippe Blain, French volleyball player and coach * Philippe Najib Boulos (1902–1979), Lebanese lawyer and politician * Philippe Coutinho, Brazilian footballer * Philippe Daverio (1949–2020), Italian art historian * Philippe Dubuisson-Lebon, Canadian football player * Philippe Ginestet (born 1954), French billionaire businessman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
